
theorem Th4:
  for L being complemented' join-commutative meet-commutative
  join-idempotent distributive upper-bounded' distributive' non empty LattStr
  for x being Element of L holds x "\/" Top' L = Top' L
proof
  let L be complemented' join-commutative meet-commutative join-idempotent
  distributive upper-bounded' distributive' non empty LattStr;
  let x be Element of L;
  x "\/" Top' L = (x "\/" Top' L) "/\" Top' L by Def2
    .= (x "\/" Top' L) "/\" (x "\/" (x `# )) by Th2
    .= x "\/" (Top' L "/\" x`# ) by Def5
    .= x "\/" x `# by Def2
    .= Top' L by Th2;
  hence thesis;
end;
